Kindly comments on Anticipating critical transitions - Less Wrong

17 Post author: PhilGoetz 09 June 2013 04:28PM

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Comment author: Kindly 10 June 2013 05:40:46PM 1 point [-]

Similar weird things happen for the Cauchy distribution (whose probability density function is proportional to 1/(1+x^2)), which is symmetric around 0 but does not have mean 0 because the sum doesn't converge.

Exercise: what do you expect to happen if you try to find the mean of the Cauchy distribution by simulation?

Comment author: shminux 10 June 2013 06:05:02PM *  3 points [-]

what do you expect to happen if you try to find the mean of the Cauchy distribution by simulation?

From the same source:

the distribution of the sample mean will be equal to the distribution of the samples themselves; i.e., the sample mean of a large sample is no better (or worse) an estimator of x0 than any single observation from the sample.

Now I'm wondering if there is a symmetric distribution where the sample mean is a strictly worse estimator of the median than a single observation.

Comment author: Matt_Simpson 12 June 2013 08:40:53PM *  0 points [-]

Off the cuff: it's probably a random walk.

Edit: It's now pretty clear to me that's false, but plotting the ergodic means of several "chains" seems like a good way to figure it out.

Edit 2: In retrospect, I should have predicted that. If anyone is interested, I can post some R code so you can see what happens.

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